The Brazilian-born Spain striker lifts the award in his first month in English football, having moved to Stamford Bridge from Atletico Madrid this summer.
Costa, 25, scored four goals in three matches at the start of the English campaign, including individual strikes against Burnley and Leicester City, and a brace against Everton in a 6-3 victory.
The striker was criticised by Everton boss Roberto Martinez following the game at Goodison on August 30 for shouting at Seamus Coleman after the Irish right-back had scored an own goal.
"I think there are certain foreign players who, when they come to the Premier League, need to understand the ethics and it's a completely different culture," he said.
Nevertheless, he beat off competition from Chelsea team-mate Cesc Fabregas, Swansea City paid Nathan Dyer and Gylfi Sigurdsson and Aston Villa's Andreas Weimann.
But Swansea were successful in the manager's category, with Garry Monk named Manager of the Month after guiding the Welsh side to three wins from three.
The Swans kicked off their Premier League campaign with a 2-1 win at Manchester United, before beating Burnley 1-0 and West Brom 3-0 to climb to joint-top of the table.